Statistics and Computing

, Volume 28, Issue 1, pp 61–75 | Cite as

On the generalization of the hazard rate twisting-based simulation approach

  • Nadhir Ben RachedEmail author
  • Fatma Benkhelifa
  • Abla Kammoun
  • Mohamed-Slim Alouini
  • Raul Tempone


Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. A naive Monte Carlo simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive, especially when dealing with rare events. An alternative approach is represented by the use of variance reduction techniques, known for their efficiency in requiring less computations for achieving the same accuracy requirement. Most of these methods have thus far been proposed to deal with specific settings under which the RVs belong to particular classes of distributions. In this paper, we propose a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of RVs. The wide scope of applicability of the proposed method is mainly due to our particular way of selecting the twisting parameter. It is worth observing that this interesting feature is rarely satisfied by variance reduction algorithms whose performances were only proven under some restrictive assumptions. It comes along with a good efficiency, illustrated by some selected simulation results comparing the performance of the proposed method with some existing techniques.


Naive Monte Carlo Rare events Importance sampling Hazard rate twisting Logarithmic efficient Twisting parameter 


  1. Albert, I.O., Marshall, W.: Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Familie. Springer, New York (2007)zbMATHGoogle Scholar
  2. Asmussen, S., Glynn, P.W.: Stochastic Simulation : Algorithms and Analysis. Stochastic Modelling and Applied Probability. Springer, New York (2007)zbMATHGoogle Scholar
  3. Asmussen, S., Kroese, D.P.: Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Probab. 38(2), 545–558 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Asmussen, S., Kortschak, D.: Error rates and improved algorithms for rare event simulation with heavy Weibull tails. Methodol. Comput. Appl. Probab. 17(2), 441–461 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Asmussen, S., Blanchet, J.H., Juneja, S., Rojas-Nandayapa, L.: Efficient simulation of tail probabilities of sums of correlated Lognormals. Ann. OR 189(1), 5–23 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Babich, F., Lombardi, G.: Statistical analysis and characterization of the indoor propagation channel. IEEE Trans. Commun. 48(3), 455–464 (2000)CrossRefGoogle Scholar
  7. Beaulieu, N.C., Rajwani, F.: Highly accurate simple closed-form approximations to Lognormal sum distributions and densities. IEEE Commun. Lett. 8(12), 709–711 (2004)CrossRefGoogle Scholar
  8. Beaulieu, N.C., Xie, Q.: An optimal Lognormal approximation to Lognormal sum distributions. IEEE Trans. Veh. Technol. 53(2), 479–489 (2004)CrossRefGoogle Scholar
  9. Ben Letaief, K.: Performance analysis of digital lightwave systems using efficient computer simulation techniques. IEEE Trans. Commun. 43(234), 240–251 (1995)CrossRefzbMATHGoogle Scholar
  10. Ben Rached, N., Benkhelifa, F., Alouini, M.-S., Tempone, R.: A fast simulation method for the Log-normal sum distribution using a hazard rate twisting technique. In: Proceedings of the IEEE International Conference on Communications (ICC’2015), London (2015a)Google Scholar
  11. Ben Rached, N., Kammoun, A., Alouini, M.-S., Tempone, R.: An improved hazard rate twisting approach for the statistic of the sum of subexponential variates. IEEE Commun. Lett. 19(1), 14–17 (2015b)CrossRefGoogle Scholar
  12. Blanchet, J., Liu, J.C.: State-dependent importance sampling for regularly varying random walks. Adv. Appl. Probab. 40(4), 1104–1128 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  14. Bucklew, J.A.: Introduction to Rare Event Simulation. Springer Series in Statistics. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  15. Chan, J.C., Kroese, D.: Rare-event probability estimation with conditional Monte Carlo. Ann. Oper. Res. 189(1), 43–61 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Devroye, L.: Non-uniform Random Variate Generation. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  17. Dupuis, P., Leder, K., Wang, H.: Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17(3), 14 (2007)CrossRefGoogle Scholar
  18. Fenton, L.: The sum of Log-normal probability distributions in scatter transmission systems. IRE Trans. Commun. Syst. 8(1), 57–67 (1960)CrossRefGoogle Scholar
  19. Filho, J.C.S.S., Yacoub, M.D.: Simple precise approximations to Weibull sums. IEEE Commun. Lett. 10(8), 614–616 (2006)CrossRefGoogle Scholar
  20. Ghavami, M., Kohno, R., Michael, L.: Ultra Wideband Signals and Systems in Communication Engineering. Wiley, Chichester (2004)CrossRefGoogle Scholar
  21. Hartinger, J., Kortschak, D.: On the efficiency of the Asmussen–Kroese-estimator and its application to stop-loss transforms. Blätter der DGVFM 30(2), 363–377 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Healey, A., Bianchi, C.H., Sivaprasad, K.: Wideband outdoor channel sounding at 2.4 GHz. In: Proceedings of the IEEE Conference on Antennas and Propagation for Wireless Communications, Waltham (2000)Google Scholar
  23. Hu, J., Beaulieu, N.C.: Accurate simple closed-form approximations to Rayleigh sum distributions and densities. IEEE Commun. Lett. 9(2), 109–111 (2005)CrossRefGoogle Scholar
  24. Jelenkovic, P., Momcilovic, P.: Resource sharing with subexponential distributions. In: Proceedings of the IEEE 21st Annual Joint Conference of the Computer and Communications (INFOCOM’ 2002), New York (2002)Google Scholar
  25. Juneja, S.: Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error. Queueing Syst. 57(2–3), 115–127 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Juneja, S., Shahabuddin, P.: Simulating heavy tailed processes using delayed hazard rate twisting. ACM Trans. Model. Comput. Simul. 12(2), 94–118 (2002)CrossRefGoogle Scholar
  27. Kroese, D.P., Rubinstein, R.Y.: The transform likelihood ratio method for rare event simulation with heavy tails. Queueing Syst. 46(3), 317–351 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kroese, D.P., Taimre, T., Botev, Z.I.: Handbook of Monte Carlo Methods. Wiley, New York (2011)CrossRefzbMATHGoogle Scholar
  29. Nandayapa, L.R.: Risk Probabilities: Asymptotics and Simulation, P.hd. thesis, university of aarhus (2008)Google Scholar
  30. Navidpour, S.M., Uysal, M., Kavehrad, M.: BER performance of free-space optical transmission with spatial diversity. IEEE Trans. Wirel. Commun. 6(8), 2813–2819 (2007)CrossRefGoogle Scholar
  31. Rubinstein, R.Y., Kroese, D.P.: The Cross Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-carlo Simulation (Information Science and Statistics). Springer, Secaucus (2004)Google Scholar
  32. Sadowsky, J.S.: On the optimality and stability of exponential twisting in Monte Carlo estimation. IEEE Trans. Inf. Theory 39(1), 119–128 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Sadowsky, J.S., Bucklew, J.A.: On large deviations theory and asymptotically efficient Monte Carlo estimation. IEEE Trans. Inf. Theory 36(3), 579–588 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Sagias, N.C., Karagiannidis, G.K.: Gaussian class multivariate Weibull distributions: theory and applications in fading channels. IEEE Trans. Inf. Theory 51(10), 3608–3619 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Schwartz, S.C., Yeh, Y.S.: On the distribution function and moments of power sums with Lognormal component. Bell Syst. Tech. J. 61, 1441–1462 (1982)CrossRefzbMATHGoogle Scholar
  36. Simon, M.K., Alouini, M.-S.: Digital Communication over Fading Channels, 2nd edn. Wiley, New York (2004)CrossRefGoogle Scholar
  37. Stüber, G.L.: Principles of Mobile Communication, 2nd edn. Kluwer Academic Publishers, Norwell (2001)Google Scholar
  38. Yilmaz, F., Alouini, M.-S.: Sum of Weibull variates and performance of diversity systems. In: Proceedings of the International Wireless Communications and Mobile Computing Conference (IWCMC’2009), Leipzig (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Computer, Electrical and Mathematical Science and Engineering (CEMSE) DivisionKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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